Dawson–Gärtner theorem
In mathematics, the Dawson–Gärtner theorem is a result in large deviations theory. Heuristically speaking, the Dawson–Gärtner theorem allows one to transport a large deviation principle on a “smaller” topological space to a “larger” one.
Statement of the theorem
Let (Yj)j∈J be a projective system of Hausdorff topological spaces with maps pij : Yj → Yi. Let X be the projective limit (also known as the inverse limit) of the system (Yj, pij)i,j∈J, i.e.
- [math]\displaystyle{ X = \varprojlim_{j \in J} Y_{j} = \left\{ \left. y = (y_{j})_{j \in J} \in Y = \prod_{j \in J} Y_{j} \right| i \lt j \implies y_{i} = p_{ij} (y_{j}) \right\}. }[/math]
Let (με)ε>0 be a family of probability measures on X. Assume that, for each j ∈ J, the push-forward measures (pj∗με)ε>0 on Yj satisfy the large deviation principle with good rate function Ij : Yj → R ∪ {+∞}. Then the family (με)ε>0 satisfies the large deviation principle on X with good rate function I : X → R ∪ {+∞} given by
- [math]\displaystyle{ I(x) = \sup_{j \in J} I_{j}(p_{j}(x)). }[/math]
References
- Dembo, Amir; Zeitouni, Ofer (1998). Large deviations techniques and applications. Applications of Mathematics (New York) 38 (Second ed.). New York: Springer-Verlag. pp. xvi+396. ISBN 0-387-98406-2. (See theorem 4.6.1)
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